Inverse Electromagnetic Scattering Problems for Long Tubular Objects

We consider the inverse time-harmonic electromagnetic scattering problem of reconstructing an object from knowledge of the generated far-field pattern for one incident field in the case of a long tubular object. Both perfectly conducting and penetrable objects are considered. The inverse scattering problem can be formulated as a nonlinear, ill-posed operator equation where the operator is the far-field map that maps the boundary of the scatterer to the far-field pattern of the scattered field. The shape of the scatterer is reconstructed using a Gauss–Newton minimization procedure for the regularized relative residual of this equation. Our main theoretical result is a characterization of the domain derivative of the far-field map for the class of tubular objects considered. Numerical examples are provided in which the computation of the electromagnetic scattered fields and their domain derivatives is carried out using boundary element methods. Even for noisy data, we obtain very accurate reconstructions of scatterers with rather complicated shapes.